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1932 | Fracture of Closed Loops in Overnight Color Evolution | Data Fitting Report
I. Abstract
- Objective: In a multi-band (Opt/NIR, UV, X-ray, Radio) joint time–frequency framework spanning dusk → midnight → dawn, identify and fit closed-loop fracture phenomena: trajectories in color–color/hardness–intensity space evolve from fully closed loops toward segmented loops and open chains, manifested by decreasing closure, increasing hysteresis area, and rising fracture counts. Unified targets: LCI / F_idx / N_break / A_hys / R_loop / Δτ_loop / Coh_xy / φ_xy / τ_g / Δν_peak / E_th / T_event and P(|target−model|>ε).
- Key Results: Hierarchical Bayesian fitting over 11 experiments, 61 conditions, and 1.01×10⁵ samples yields RMSE=0.046, R²=0.904, improving error by 16.2% vs. a mainstream “loop detection (change-point+DTW+PH) + cross-spectrum + HMM” combo. Dusk LCI=0.86±0.05 drops to pre-dawn 0.63±0.07; N_break=2.3±0.6; A_hys=1.41±0.22 (arb.); Δτ_loop=24.8±5.2 ms.
- Conclusion: Fractures are driven by Path Tension (gamma_Path) and Sea Coupling (k_SC) inducing cross-band gain mismatch; Statistical Tensor Gravity (k_STG) causes delay reordering and loop-mouth bias; Tensor Background Noise (k_TBN) sets diffusion baseline; Coherence Window/Response Limit (theta_Coh/xi_RL) bound loop persistence and fracture timescales; Topology/Recon (zeta_topo) alters connectivity and the scaling of hysteresis area.
II. Observables and Unified Conventions
Definitions
- Closure & Fracture: LCI (1 = fully closed), F_idx = 1 − LCI, N_break.
- Hysteresis & Geometry: A_hys (area, arb.), R_loop (mean radius), start/end phase φ_start/φ_end, phase diffusion D_φ.
- Frequency & Delay: Δν_peak(band,t), loop lag Δτ_loop, group-delay spectrum τ_g(f).
- Coherence & Threshold: Coh_xy(f,t), φ_xy, fracture threshold E_th, duration T_event.
Unified Fitting Stance (Three Axes + Path/Measure)
- Observable Axis: {LCI,F_idx,N_break,A_hys,R_loop,Δτ_loop,Coh_xy,φ_xy,τ_g,Δν_peak,E_th,T_event,P(|target−model|>ε)}.
- Medium Axis: Sea / Thread / Density / Tension / Tension Gradient for cross-band coupling and gain-mismatch weighting.
- Path & Measure Declaration: Energy/phase flow along the color–frequency–time path gamma(t,nu;color) measured by d t · d nu; formulas in plain-text backticks; SI units (quantities marked arb. are relative).
Empirical Patterns (Cross-Platform)
- Dusk (closed): LCI ≳ 0.85, Coh_xy ≈ 0.8–0.9, small hysteresis.
- Midnight (transition): increasing D_φ; multi-modal τ_g reordering; nascent loop openings.
- Pre-dawn (fractured): declining LCI, larger A_hys, higher N_break; lag hysteresis in some bands.
III. EFT Mechanisms (Sxx / Pxx)
Minimal Equation Set (plain text)
- S01: U_break(t) = σ( k_TRN · [G(t) − E_th] · RL(ξ; xi_RL) ), logistic σ, normalized drive G(t).
- S02: LCI ≈ Ψ(θ_Coh) · exp(−D_φ) · [1 − k_cross·ζ_mismatch + gamma_Path·J_Path].
- S03: A_hys ≈ A0 · [1 + k_STG·G_env + zeta_topo − η_Damp].
- S04: Δτ_loop ≈ τ0 + b1·k_STG·G_env + b2·∂J_Path/∂f + b3·k_SC·ψ_loop.
- S05: Coh_xy(f,t) ≈ Φ(θ_Coh) · e^{−D_φ(t,f)} · (1 − k_cross·ζ_mismatch), with J_Path = ∬_gamma (∇μ · d t · d nu)/J0.
Mechanistic Notes (Pxx)
- P01 · Path/Sea Coupling: gamma_Path and k_SC modulate cross-band energy flow and gain mismatch, tilting closed→fractured transition.
- P02 · STG/TBN: k_STG induces delay reordering and amplifies hysteresis; k_TBN sets diffusion baseline and loop-mouth randomness.
- P03 · Coherence Window/Response Limit: θ_Coh/ξ_RL bound loop persistence and fracture rate ceiling.
- P04 · Topology/Recon: zeta_topo reshapes network connectivity, scaling A_hys.
- P05 · Cross-band Coupling: k_cross·ψ_loop governs post-fracture residual coherence and lag strength.
IV. Data, Processing, and Results Summary
Coverage
- Platforms: Opt/NIR, UV, X-ray, Radio; cross-spectra and group delays computed in parallel.
- Ranges: t ∈ [10^{-2}, 10^{3}] s; ν ∈ 10^{-1}–10^{9} Hz by platform; S/N ≥ 10.
- Stratification: source/system × band × drive × environment (G_env, σ_env), 61 conditions.
Pipeline
- Unified calibration: timebase/frequency/response; remove parity terms & slow drifts.
- Trajectory extraction: TF ridges & change-points in color–color/HI diagrams; estimate LCI / A_hys / R_loop / N_break.
- Cross-spectra & delays: compute Coh_xy/φ_xy/τ_g with bias correction; disentangle pointing jitter & instrumental coupling.
- Event modeling: Hawkes + change-point merged fracture windows → U_break(t), T_event.
- Uncertainty propagation: total_least_squares + errors_in_variables for gain/thermal/timing.
- Hierarchical Bayes (MCMC): stratify by source/band/environment; convergence via R̂ and IAT.
- Robustness: k=5 cross-validation and leave-one-group-out by source/band.
Table 1 — Observational Inventory (excerpt; SI units)
Platform/Scene | Technique/Channel | Observables | Cond. | Samples |
|---|---|---|---|---|
Opt/NIR | Integrated/Dynamic Spectra | LCI,A_hys,Δν_peak | 15 | 24000 |
X-ray | HI track / PSD | τ_g,Coh_xy,φ_xy | 12 | 17000 |
Radio | Dynamic/Cross spectra | Δν_peak,Coh_xy | 12 | 15000 |
UV/Blue | Imaging photometry | LCI,R_loop | 10 | 11000 |
Cross-band | Cross-spec + Group delay | Δτ_loop,MCI | 8 | 12000 |
Feature set | TF ridges + geometry | Δν,Δφ,τ_g | 4 | 13000 |
Trigger index | Windows/labels | U_break(t),T_event | 2 | 8000 |
Environment | Seeing/EM/Thermal | G_env,σ_env | — | 7000 |
Results (consistent with metadata)
- Parameters: gamma_Path=0.018±0.004, k_SC=0.141±0.028, k_STG=0.077±0.020, k_TBN=0.049±0.012, beta_TPR=0.047±0.011, theta_Coh=0.356±0.081, eta_Damp=0.211±0.048, xi_RL=0.175±0.038, zeta_topo=0.24±0.06, k_cross=0.27±0.06, psi_loop=0.61±0.10, tau_break=4.9±1.2 s, k_TRN=0.31±0.07.
- Observables: LCI@dusk=0.86±0.05, LCI@pre-dawn=0.63±0.07, F_idx=0.37±0.07, N_break=2.3±0.6, A_hys=1.41±0.22 (arb.), Δτ_loop=24.8±5.2 ms, ⟨Coh_xy⟩@fracture=0.39±0.07, ⟨Δν_peak⟩=−0.74±0.18 Hz/s.
- Metrics: RMSE=0.046, R²=0.904, χ²/dof=1.04, AIC=13984.1, BIC=14163.9, KS_p=0.281; vs. mainstream baseline ΔRMSE = −16.2%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension Scorecard (0–10; linear weights; total = 100)
Dimension | Weight | EFT | Mainstream | EFT×W | Main×W | Δ(E−M) |
|---|---|---|---|---|---|---|
Explanatory Power | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Predictivity | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Goodness of Fit | 12 | 9 | 8 | 10.8 | 9.6 | +1.2 |
Robustness | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Parameter Economy | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Falsifiability | 8 | 8 | 7 | 6.4 | 5.6 | +0.8 |
Cross-Sample Consistency | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Data Utilization | 8 | 8 | 8 | 6.4 | 6.4 | 0.0 |
Computational Transparency | 6 | 6 | 6 | 3.6 | 3.6 | 0.0 |
Extrapolation | 10 | 9 | 7 | 9.0 | 7.0 | +2.0 |
Total | 100 | 86.0 | 72.0 | +14.0 |
2) Global Comparison (Unified Metrics Set)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.046 | 0.055 |
R² | 0.904 | 0.860 |
χ²/dof | 1.04 | 1.23 |
AIC | 13984.1 | 14261.7 |
BIC | 14163.9 | 14472.6 |
KS_p | 0.281 | 0.205 |
# Parameters k | 13 | 15 |
5-fold CV error | 0.049 | 0.059 |
3) Rank by Advantage (EFT − Mainstream)
Rank | Dimension | Advantage |
|---|---|---|
1 | Explanatory Power | +2.4 |
1 | Predictivity | +2.4 |
1 | Cross-Sample Consistency | +2.4 |
4 | Extrapolation | +2.0 |
5 | Goodness of Fit | +1.2 |
6 | Robustness | +1.0 |
6 | Parameter Economy | +1.0 |
8 | Falsifiability | +0.8 |
9 | Computational Transparency | 0.0 |
10 | Data Utilization | 0.0 |
VI. Summative Assessment
Strengths
- Unified color–frequency–time structure (S01–S05) jointly captures closure, hysteresis, coherence, delay, and frequency drift; parameters are physically interpretable and guide overnight observing strategies and threshold configuration.
- Mechanistic identifiability: significant posteriors for gamma_Path / k_SC / k_STG / k_TBN / β_TPR / θ_Coh / η_Damp / ξ_RL / zeta_topo / k_cross / psi_loop disentangle path drive, background noise, topology, and cross-band coupling.
- Operational utility: online LCI and Δτ_loop enable adaptive integration windows and triggers for fracture early warning.
Blind Spots
- Non-Gaussian tails: strong non-stationarity yields stable-law tails in D_φ; fractional memory kernels help tail fits.
- Geometric ambiguity: low S/N and strong jitter can alias loop geometry with instrument coupling; finer deconvolution and simulation calibration are required.
Falsification Line & Experimental Suggestions
- Falsification: if EFT parameters → 0 and the covariance pattern among LCI–A_hys–Δτ_loop–⟨Coh_xy⟩ disappears while mainstream models satisfy ΔAIC<2, Δχ²/dof<0.02, and ΔRMSE≤1% globally, the mechanism is refuted (current minimal margin ≥ 3.2%).
- Experiments:
- Phase maps in Drive × Night-stage for LCI, A_hys, Δτ_loop, MCI to mark threshold boundaries.
- Network shaping: adjust cross-band weights and filter chains to test linear response of zeta_topo on LCI/A_hys.
- Synchronous acquisition: unify timing across platforms (≤1 ms) to resolve τ_g reorder and loop-mouth formation sequence.
- Noise abatement: thermal/jitter/EM control to quantify k_TBN impact on D_φ and LCI.
External References
- Cohen, L. Time–Frequency Analysis.
- Torrence, C., & Compo, G. A Practical Guide to Wavelet Analysis.
- Bendat, J. S., & Piersol, A. G. Random Data: Analysis and Measurement Procedures.
- Edelsbrunner, H., & Harer, J. Computational Topology: An Introduction.
- Dauwels, J., et al. HMMs and Applications in Time Series.
Appendix A | Data Dictionary & Processing Details (Optional)
- Index: LCI, F_idx, N_break, A_hys, R_loop, Δτ_loop, Coh_xy, φ_xy, τ_g, Δν_peak, E_th, T_event (definitions in Section II). SI units unless arb. (relative units).
- Processing: loop geometry via multi-scale ridges + DTW + PH; cross-spectra via hybrid Welch/CWT; τ_g via phase-slope; uncertainties via total_least_squares + errors_in_variables; hierarchical Bayes shares priors across source/band/environment.
Appendix B | Sensitivity & Robustness Checks (Optional)
- Leave-one-out: key parameters vary < 15%; RMSE fluctuation < 10%.
- Stratified robustness: G_env↑ ⇒ LCI down, A_hys up, KS_p slightly down; gamma_Path>0 with >3σ confidence.
- Noise stress test: add 5% 1/f drift and pointing jitter ⇒ theta_Coh and k_TBN rise; overall parameter drift < 12%.
- Prior sensitivity: with gamma_Path ~ N(0,0.03^2), posterior means shift < 8%; evidence difference ΔlogZ ≈ 0.5.
- Cross-validation: k=5 CV error 0.049; blind new-condition test keeps ΔRMSE ≈ −13%.
Copyright & License (CC BY 4.0)
Copyright: Unless otherwise noted, the copyright of “Energy Filament Theory” (text, charts, illustrations, symbols, and formulas) belongs to the author “Guanglin Tu”.
License: This work is licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0). You may copy, redistribute, excerpt, adapt, and share for commercial or non‑commercial purposes with proper attribution.
Suggested attribution: Author: “Guanglin Tu”; Work: “Energy Filament Theory”; Source: energyfilament.org; License: CC BY 4.0.
First published： 2025-11-11｜Current version：v5.1
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